Exact Solutions of the Schrödinger Equation for Pseudoharmonic Potential

The pseudoharmonic potential belongs to the small number of problems for which the Schrödinger equation is exactly solvable [1] and is thus of intrinsic interest. The solution is outlined in the Details below.
For selected parameters and , you can either display an energy diagram, showing the first several eigenvalues superposed on the potential energy curve, or a plot of the radial function for a selected value of the quantum number .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The Schrödinger equation for the radial function , in the case of zero angular momentum, is given by
.
The eigenfunctions for bound states are found to be
,
where is an associated Laguerre polynomial. The solution of the differential equation initially gave associated Laguerre functions , with
and
.
The quantization is determined by the condition that the eigenfunctions must approach 0 as . The asymptotic behavior of Laguerre functions is given by:
as .
This would overtake the converging factor in the solution as , unless is an integer , which would then produce a singularity in a gamma function of the denominator. The corresponding eigenvalues are thereby determined:
with .
Reference
[1] I. I. Gol'dman and V. D. Krivchenkov, Problems in Quantum Mechanics (B. T. Geǐlikman, ed., E. Marquit and E. Lepa, trans.), Reading, MA: Addison-Wesley, 1961.‬
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.